List of harmonic analysis topics explained

This is a list of harmonic analysis topics. See also list of Fourier analysis topics and list of Fourier-related transforms, which are more directed towards the classical Fourier series and Fourier transform of mathematical analysis, mathematical physics and engineering.

Fourier analysis

Fourier series

• Periodic function
• Trigonometric function
• Trigonometric polynomial
  • Exponential sum
• Dirichlet kernel
• Fejér kernel
• Gibbs phenomenon
• Parseval's identity
• Parseval's theorem
• Weyl differintegral
• Generalized Fourier series
  • Orthogonal functions
  • Orthogonal polynomials
  • Empirical orthogonal functions
• Set of uniqueness

Fourier transform

• Continuous Fourier transform
• Fourier inversion theorem
• Plancherel's theorem
• Convolution
• Convolution theorem
• Positive-definite function
• Poisson summation formula
• Paley-Wiener theorem
• Sobolev space
• Time–frequency representation
• Quantum Fourier transform

Topological groups

• Topological abelian group
• Haar measure
• Discrete Fourier transform
  • Dirichlet character
• Amenable group
  • Von Neumann's conjecture
• Pontryagin duality
• Kronecker's theorem on diophantine approximation
• Almost periodic function
• Bohr compactification
• Wiener's tauberian theorem

Representation theory

See main article: List of representation theory topics.

• Representation of a Lie group
• Unitary representation
• Irreducible representation
• Harish-Chandra character
• Restricted representation
• Induced representation
• Peter–Weyl theorem
• Spherical harmonic
• Casimir operator
• Hecke operator
• Stone–von Neumann theorem
• Discrete series representation
• Tempered representation
• Langlands program

Fast Fourier transform

• Bluestein's FFT algorithm
• Cooley–Tukey FFT algorithm
• Rader's FFT algorithm
• Number-theoretic transform
• Irrational base discrete weighted transform

Analysis of unevenly spaced data

• Least-squares spectral analysis

Applications

• FFT multiplication
• Spectral method
• Fourier transform spectroscopy
• Signal analysis
• Analytic signal
• Welch method